This might not be a SharePoint specific issue.
I need to access a non-SharePoint web service from a SharePoint application page. The web service is accessed through two calls, one with a username and password that returns a token, and the second uses that token and returns JSON. The first call uses https, the second http. Same server, different service, I believe. This server is on a different domain than the SharePoint server.
The call fails with "The underlying connection was closed: Could not establish trust relationship for the SSL/TLS secure channel". I am able to complete both calls through IE on the SharePoint server.
Edit - this is a SharePoint specific issue - I created a little console app that does the same thing, and it works just fine running on the SharePoint server.